Solving the walk

To determine the solution to the set of ODEs defined in equation (1), we employ a straightforward Euler scheme. Time is discretized into steps of \(\Delta t\), and the evolution is expressed as:

\[ \begin{align}\begin{aligned}x_{i, n + 1} = x_{i, n} + \Delta t \, \dot{x}_{i,n}\\\dot{x}_{i, n + 1} = \dot{x}_{i, n} + \ddot{x}_{i, n} \, \Delta t \, .\end{aligned}\end{align} \]

In these equations, \(x_{i, n}\) represents the position of walker \(i\) at time \(t = n \Delta t\), and a similar notation is employed for velocity and acceleration. The acceleration, of course, is computed using equation (1).

In our library, these calculations are carried out by the Solver class, which simplifies the process significantly. For instance, if we aim to find the solution at a specific time, say \(t = 1\) second, the code might look like this:

import mockWalkers as mckw

# Define the problem ...
# walkers = ...
# delta_t = ...
# ...

s = mckw.Solver(
    walkers,
    delta_t,
    vd_calcs,
    obstacles,
    geometry,
)

while s.current_time < 1:
    s.iterate()

# Do something with the solution ...

The various elements used for describing the problem will be explained in detail later. What’s crucial to note here is the Solver.iterate method. Each time this method is called, it performs one iteration of the Euler scheme, advancing the state from \(t = n \Delta t\) to \(t = (n + 1) \Delta t\). This approach simplifies the numerical solution of the ODEs, allowing you to obtain the desired solution at any desired time point.